Exercises: Stability of systems

Explain in your own words what it means for an automatic system to be stable and give two examples: one of a stable system (a pendulum with friction) and one of an unstable system (a broom balanced vertically on the palm of a hand).

Distinguish between the two states in the time response of a system: transient state and steady state. In the example of a heater being turned on, which part of the temperature curve corresponds to each?

The figure shows the response of three systems to the same step input. Classify each response (A, B and C) as stable overdamped, stable underdamped (damped oscillatory) or unstable, and justify your choice based on how each curve evolves over time.

Define the steady-state error of a control system and explain, using a heating system as an example, what it means for it to be non-zero (the room settles half a degree below the setpoint). What type of controller eliminates it completely?

Explain what step and ramp test inputs are and what information each provides about a system (step: speed, overshoot and steady-state error; ramp: ability to track setpoints that change over time). Give a real-world example of each type of setpoint.

A first-order system responds by rising gradually without oscillations, like the temperature of a heater. Explain what its time constant $\tau$ is (time to reach $63\,\%$ of the total change) and calculate how long a system with $\tau = 20\ \text{s}$ takes to practically reach its final value (about $4\tau$ to $5\tau$).

Compare the stability of a ball at the bottom of a bowl, a ball on top of a dome, and a ball on a flat horizontal surface. Relate each case to a stable system, an unstable system, and a critically stable system (it neither returns nor rolls away), and give a technological example of each.

Explain what a sustained oscillation is (it neither grows nor dies away) and why it is the boundary between a stable and an unstable system. In which devices is it intentionally desired (electronic oscillators, clocks)?

A well-tuned thermostat brings the room to $21\ ^\circ\text{C}$ without oscillations; a poorly tuned one alternates between heating and cooling cycles. Relate each behaviour to an overdamped or underdamped response and explain which controller parameter should be adjusted in the second case.

Explain what overshoot and settling time are in a step response, and comment on the usual trade-off between speed and overshoot: why does a very fast system tend to oscillate more?

A data centre uses simulators to test its control systems before installing them. Explain the advantages of experimenting with simulation (safety, cost, repeatability, ability to force extreme situations) and what fundamental limitation a simulated model has.

The figure shows the response of a system to a step of $1$ unit. a) Read the final value and peak value from the graph and calculate the percentage overshoot. b) Estimate the settling time (band of $\pm 5\,\%$ of the final value). c) Calculate the steady-state error.

Two proportional controllers for the same heater use gains $K = 2$ and $K = 10$. Predict and explain the behaviour of each: speed, overshoot and steady-state error, and justify why an intermediate gain is usually the practical choice.

The operator of a thermal power plant takes $5\ \text{s}$ to notice any change (pure measurement delay). Explain why pure delay is the worst enemy of a control loop (the correction arrives when the error has already changed) and what consequences it has on the maximum usable gain before oscillations appear.

Find examples of deliberately unstable systems or systems in sustained oscillation that are useful: the oscillator of a digital watch, the multivibrator of a car indicator, the acoustic feedback howl at a concert. Explain for each why it is designed this way and what prevents it from 'blowing up' (non-linearities, saturation).

In a simulator, the step response of a system takes $2\ \text{s}$ to reach $63\,\%$ of the total change and does not oscillate. a) Estimate its time constant and its first-order model. b) Predict its output at $4\ \text{s}$ and $10\ \text{s}$ after a step of $100\,\%$. c) Explain how you would verify in the simulator that it is indeed first order.

A prototype of a self-balancing two-wheeled transport vehicle (Segway type) uses a gyroscope and accelerometers to keep the mast vertical by acting on the motor torque at the wheels (a system that can be modeled as an inverted pendulum).

During closed-loop calibration tests, three anomalous situations are observed:

1. a) With a very low proportional gain ($K_p = 4$), upon an external step disturbance (a push) the vehicle barely reacts and slowly falls to the ground.
2. b) When increasing the gain to a very high value ($K_p = 85$) seeking an energetic response, the vehicle does not stabilize: it enters a violent oscillation of increasing frequency and ends up tipping over.
3. c) A developer proposes solving the oscillation in b) by introducing a digital filter into the sensor readings that adds a time delay of $120\text{ ms}$.

Physically justify the behavior of a) and b) in terms of dynamics and closed-loop stability. Predict with reasoning the effect that the developer's proposal will have on the stability of the vehicle and propose which basic control action (integral or derivative) you would add to damp the oscillations without destabilizing the system, justifying your choice.

During flight tests of a quadcopter drone on a test bench, the roll angle (horizontal tilt) stabilization loop is subjected to a sudden gust of lateral wind acting as a constant disturbance. The control team tests three different tunings of the proportional controller:

• Configuration A: The drone tilts $14^\circ$ and remains banked at that constant angle while the wind lasts, without recovering the horizontal.

• Configuration B: After the wind, the drone oscillates indefinitely around the horizontal with a pure sinusoidal oscillation of constant amplitude of $\pm 6^\circ$ at $3\text{ Hz}$ that neither grows nor damps out.

• Configuration C: The drone begins to oscillate with increasing amplitudes ($\pm 3^\circ, \pm 8^\circ, \pm 22^\circ\dots$) until the safety system disconnects the motors to prevent damage.

Interpret the three behaviors by relating each case to the loop gain and the system stability condition (asymptotically stable with error, at the boundary of stability, or unstable). Justify what physical limitation of the P controller causes the failure in configuration A and what gain adjustment you must make in configuration C to return the system to the stable region.

In the level control of a cylindrical chemical reagent tank, the setpoint changes abruptly from $1{,}5\text{ m}$ to $2{,}5\text{ m}$. The technical safety and quality requirements mandate:

1. Maximum overshoot equal to zero or below $8\,\%$ (to prevent overflow over the crest located at $2{,}75\text{ m}$).

2. Settling time ($t_s$ at $\pm 2\,\%$) under $20\text{ s}$.

3. Steady-state level error equal to zero ($e_{rp} = 0$).

Two proposals are evaluated in the simulator:

• Proposal 1 (P controller with $K_p = 12$): The level rises monotonically without oscillation, settling after $11\text{ s}$ at a constant value of $2{,}28\text{ m}$.

• Proposal 2 (PI controller with $K_p = 12$ and $T_i = 3\text{ s}$): The level reaches a maximum peak of $2{,}82\text{ m}$ at $8\text{ s}$, undergoes three damped oscillations, and settles exactly at $2{,}50\text{ m}$ after $34\text{ s}$.

1. a) Numerically evaluate compliance with the three requirements for each proposal and determine whether either one is valid.
2. b) Justify the physical/mathematical cause of the failure of Proposal 1 and the failure of Proposal 2.
3. c) Design a specific modification of the parameters ($K_p$ and/or $T_i$) for Proposal 2 that makes it possible to meet all requirements simultaneously, reasoning how your changes will affect the overshoot and the settling time.

The ventilation system of a drying room uses an exhaust fan driven by a relay with on-off control with hysteresis to regulate relative humidity. The room generates water vapor at a constant rate, and the exhaust fan removes a fixed flow rate when it is on.

1. a) In the initial configuration, the switching thresholds are set at $66\,\%$ (turned on) and $64\,\%$ (turned off). The relay switches 45 times per hour, causing premature breakdowns due to arcing and electromechanical fatigue. Justify why a narrow hysteresis margin increases the switching frequency and predict what would happen if an operator completely eliminated hysteresis by setting a single cutoff threshold at $65\,\%$.
2. b) A technician proposes widening the band to $75\,\%$ (turned on) and $50\,\%$ (turned off) to protect the relay. Analyze the impact this measure will have on the drying process and on the room's thermal energy consumption.
3. c) Justify technically and energetically what advantages replacing the relay-exhaust fan assembly with a fan with a variable frequency drive and proportional control would provide, and predict whether with this new system the relative humidity will remain exactly at the setpoint if steam generation in the room suddenly doubles.

In a large-volume automated greenhouse, the motorized opening of the roof vents takes $3\text{ minutes}$ to replace the hot air, and the temperature sensor is placed in the center of the crop at ground level, introducing an additional transport dead time of $2\text{ minutes}$ into the measurement.

The system is controlled by a proportional controller that actuates the opening based on the internal temperature.

1. a) During sunny days with variable breeze, the system enters a continuous high-amplitude thermal oscillation cycle (the vents open to $100\,\%$, the crop cools down excessively, the vents close completely, the temperature rises rapidly, and the cycle repeats indefinitely). Explain the cause of this instability by analyzing the interaction between the controller gain and the total delay (dead time + plant time constant).
2. b) Reasonably refute the proposal of a supervisor who suggests: "To eliminate these oscillations and stabilize the system at the setpoint, we must triple the controller gain so that the vents react with maximum speed as soon as the temperature varies by a tenth of a degree."
3. c) Design a control solution for this problem: justify how you would readjust the internal temperature loop and explain how incorporating a pyranometer (external solar radiation sensor) into a feedforward control structure makes it possible to prevent overheating before the internal delay occurs.

An industrial electric furnace is governed by a PID control loop to reach and maintain a setpoint temperature of $750^\circ\text{C}$. The power stage of the heating resistors is limited to a maximum physical saturation of $100\,\%$ ($40\text{ kW}$).

When the furnace starts up from room temperature ($20^\circ\text{C}$), the control signal immediately demands $100\,\%$ of the power. However, once the temperature finally reaches $750^\circ\text{C}$, the actuator keeps delivering $100\,\%$ of the power for another $8\text{ minutes}$, which causes the temperature to shoot up to $890^\circ\text{C}$ (severe overshoot) and take more than half an hour to settle.

1. a) Identify the phenomenon responsible for this delay in cutting the power, justifying what happens to the numerical value accumulated in the integral action (which adds up the error over time) while the furnace heats from $20^\circ\text{C}$ to $750^\circ\text{C}$ with the actuator at its maximum.
2. b) Justify why a purely proportional control would not show this time delay when switching off, but explain what unacceptable permanent error it would leave in the furnace in steady state.
3. c) Propose and describe two technical modifications (one in the configuration of the control algorithm and another in the parameter tuning process) to avoid overheating during start-ups while maintaining zero error in steady state.

In the sound system of a conference room, when the amplifier's volume potentiometer is turned up past a certain level, acoustic feedback (the Larsen effect) occurs suddenly, manifested as a deafening high-pitched squeal at a fixed frequency of $3{,}2\text{ kHz}$ that saturates the loudspeakers.

1. a) Qualitatively model this system as a closed loop using a block diagram, clearly identifying the reference input (voice), the acoustic summing junction, the microphone, the amplifier, the loudspeaker, and the transfer function of the airborne medium (feedback path).
2. b) Formally justify the appearance of the squeal using the closed-loop oscillation criterion (conditions of loop gain $|A \cdot \beta|$ and total phase shift $\phi$). Reasonably explain why the system oscillates precisely at $3{,}2\text{ kHz}$ and does not generate white noise spanning the entire audible spectrum at once.
3. c) Justify, based on closed-loop stability theory, why each of the following three technical interventions eliminates the problem: moving the microphone away from the line of the loudspeakers, using a unidirectional (cardioid) microphone pointing away from the speakers, and inserting a notch filter tuned to $3{,}2\text{ kHz}$.

In the speed control tuning of a roller in a paper rolling line, the closed-loop response to a setpoint step of $600\text{ rpm}$ is tested in a simulator by modifying only the proportional gain $K_p$. The following records are obtained:

1. a) If the technical specifications require a maximum overshoot below $10\,\%$ and the shortest possible settling time, numerically justify which test represents the optimal compromise.
2. b) Analyze the evolution of the data in the table: physically justify why increasing $K_p$ reduces the steady-state error, but from test 3 onward the settling time worsens instead of continuing to decrease.
3. c) The client strictly demands that the final speed be exactly $600\text{ rpm}$ ($e_{rp} = 0$) while keeping the overshoot below $10\,\%$. Demonstrate why it is impossible to satisfy this demand by varying only the gain $K_p$, and indicate what additional control action you must incorporate and what modification you should make to the $K_p$ value of the selected test when adding it.

Unit challenge. Design the thermal control of your room against actual weather fluctuations.

1. Data collection: Go to the AEMET website and record the maximum and minimum outdoor temperatures of your municipality registered in the last $24$ hours to quantify the ambient thermal disturbance.

2. Modelling and simplification: Draw the block diagram of the system composed of a proportional controller $K_p$, the thermal plant $G(s) = \frac{1,5}{s + 0,4}$, an additive disturbance, and a unit feedback sensor $H(s) = 1$. Apply the rules of block algebra to simplify the system and obtain its closed-loop transfer function $M(s)$.

3. Stability study: Analyse the stability of the system by evaluating the position of the poles of $M(s)$ (or applying the Routh-Hurwitz criterion) as a function of $K_p$. Determine the range of gains that guarantees stable operation.

4. Reasoned proposal: In LibreOffice Calc, calculate and plot the time response of the system to a thermal step based on the AEMET data for two different values of $K_p$. Justify in a short report which one you would choose to avoid overshoot.

Evaluation criteria:

• Rigour in the algebraic reduction of the block diagram.

• Correct mathematical determination of stability conditions.

• Consistency in the justification of the gain $K_p$ from the simulation and climate data.